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Cauchy sequence proof. 
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#1
Mar412, 12:57 AM

P: 2,466

1. The problem statement, all variables and given/known data
Assume [itex] x_n [/itex] and [itex] y_n [/itex] are Cauchy sequences. Give a direct argument that [itex] x_n+y_n [/itex] is Cauchy. That does not use the Cauchy criterion or the algebraic limit theorem. A sequence is Cauchy if for every [itex] \epsilon>0 [/itex] there exists an [itex] N\in \mathbb{N} [/itex] such that whenever [itex] m,n\geq N [/itex] it follows that [itex] a_na_m< \epsilon [/itex] 3. The attempt at a solution Lets call [itex] x_n+y_n=c_n [/itex] now we want to show that [itex] c_mc_n< \epsilon [/itex] Lets assume for the sake of contradiction that [itex] c_mc_n> \epsilon [/itex] so we would have [itex]x_m+y_mx_ny_n> \epsilon [/itex] [itex] x_m> \epsilon+y_ny_m [/itex] since [itex] y_n>y_m [/itex] and we know that [itex] x_m< \epsilon [/itex] so this is a contradiction and the original statement must be true. 


#2
Mar412, 01:08 AM

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PF Gold
P: 4,500

To get the contradiction you're going to want to use the triangle inequality on (x_{n}x_{m})+(y_{n}y_{m}) 


#3
Mar412, 03:48 PM

P: 2,466

ok thanks for your response.
So I take [itex] (x_nx_m)+(y_ny_m) \leq x_nx_m+y_ny_m [/itex] lets assume that [itex] x_nx_m+y_ny_m > \epsilon [/itex] Im going to rewrite it as [itex] A+B> \epsilon [/itex] so now we have [itex] A> \epsilon B [/itex] Can I just say this since we know that [itex] A< \epsilon [/itex] and [itex]B< \epsilon [/itex] since [itex] \epsilon [/itex] can be any number bigger than zero, then both of these values should be less than [itex] \frac{\epsilon}{2} [/itex] therefore [itex] A+B< \epsilon [/itex] I have a feeling my last step is not ok 


#4
Mar412, 03:55 PM

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PF Gold
P: 4,500

Cauchy sequence proof.



#5
Mar412, 03:58 PM

P: 2,466

Ok , Am I thinking about this in the right way.



#6
Mar412, 04:10 PM

Emeritus
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PF Gold
P: 4,500

Well, I can't read your mind but the part I quoted is the basis of the a correct argument for the proof. You just have to add the fact that these are sequences  A and B aren't always that small, but as long as n and m are big enough they are (by definition of the x's and the y's forming Cauchy sequences)



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